arXiv research

Sharp inequality in spaces with non-negative Ricci curvature.

Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.

Three-manifolds with non-negative pinched Ricci curvature have complete Ricci flows.

Compact Kähler orbifolds with non-negative Ricci curvature are simply connected.

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.

Smooth 3D flows from non-smooth starting points.

Formal manifolds with non-negative Ricci curvature have formal covers.

The paper explores curvatures on graphs and their implications for Ricci flatness.

New proof shows certain 3D spaces are essentially like infinite space.

New Bochner technique for foliations with non-negative Ricci curvature.

The paper proves conjectures and classifies metrics on 3D manifolds.

New metrics with non-negative scalar curvature are always Ricci-flat on certain surgeries.

Sharp isoperimetric inequality for Finsler manifolds with non-negative Ricci curvature.

Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.

Sharp isoperimetric inequality on Finsler manifolds with non-negative Ricci curvature.

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.

Study improves understanding of Ricci curvature in manifolds.

Study on stable minimal hypersurfaces under Ricci curvature constraints.

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than -c^2(i), where c^2(i…

Log-Sobolev inequality proven for submanifolds in specific types of manifolds.

We prove that the space of smooth Riemannian metrics on the three-ball with non-negative Ricci curvature and strictly convex boundary is path connected; and, moreover, that the associated moduli space (i.e., modulo orientation-preserving diffeomorphisms of the three-ball) is contractible. As an application, using resul…

For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.

The study uses Ricci flow to prove flatness of certain Riemannian manifolds.

The paper generalizes a Steklov eigenvalue inequality for substatic triples under non-negative Ricci curvature.

Proves uniqueness of Ricci flow with scaling invariant estimates.

Study reveals flatness of Hessian metrics with non-negative Ricci curvature on foliation leaves.

Entropy defined for submanifolds; applies to mean curvature flow limits of surfaces.

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show tha…

In this paper, we prove that the $L^p$ essential spectra of the Laplacian on functions are $[0,+\infty)$ on a non-compact complete Riemannian manifold with non-negative Ricci curvature at infinity. The similar method applies to gradient shrinking Ricci soliton, which is similar to non-compact manifold with non-negative…

We prove that any complete metric on R^3 minus a ball with non-negative Ricci curvature and quadratic Ricci-curvature decay, has cubic volume growth.

Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.

A result of M. Ledoux is that a complete Riemannian manifold with non negative Ricci curvature satisfying the Euclidean Sobolev inequality is the Euclidean space. We present a shortcut of the proof. We also give a refinement of a result of B-L. Chen et X-P. Zhu about locally conformally flat manifolds with non negative…

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly $\t…

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolve…

The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita…

In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result Böhm and Wi…

Paper proves rigidity of minimal disks in 3-balls with non-negative Ricci curvature.

Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.

Let $(M, g)$ be a complete non-compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in \cite{CT3}, we prove that the universal cover $\wt M$ of $M$ is biholomorphic to $\ce^n$ provided either that $(M, g)$ has average quadratic curvature decay, or $…

We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that it is a quotient of $Σ\times \mathbb{C}^{n-1-k}\times N^k$, where $Σ$ and $N$ denot…

The paper proves isoperimetric inequalities in manifolds with small negative Ricci curvature.

Study shows bound on Uryson width for specific 3D manifolds.

Study almost rigidity of super Ricci flow with non-negative Muller quantity.

Lower bound on minimum vertex degree for non-negative Lin-Lu-Yau curvature on graphs.